[R-SIG-Finance] Zivot vs. Engle vs. Stoffer - help with the meaning of different GARCH notations, please!

[Brian G. Peterson]

tom soyer wrote:
> Thanks Brian. Sorry about the reference being not clear. Here is my 2nd try:
>  
> Zivot: MFTSWS, 2003 edition. [1] page 212, equation 7.1, and [2] page 
> 216, equation 7.5
> Engle: Garch101, middle of page 160.
> Stoffer: you had it exactly right. page 286, equation 5.30 and 5.44.

I don't have the Engle reference handy, so my comments should be 
construed as mostly from Zivot and Wang, and Shumway and Stoffer.  The 
Engle notations appear to be equivalent, with a possible additional 
term, as I said in my prior email.

> So my follow up question to you is why do "these models all appear 
> equivalent" to you? Take equation [6] for example, it says to me that 
> one could model the variance part of the process using garch and only 
> garch. If these notations were all the same, then garch alone should 
> also be able to model [2], right? But it seems to me that that's not the 
> case, i.e., arma + garch is needed to model [2]. Do you know what I 
> mean? Does that make sense?

Eq's [1],[3],[5] in your list all refer to an AR(1) model for the 
returns, of the variance modified by a white-noise parameter.

Eq's [2],[4],[6] in your list all describe the GARCH(1,1) "generalized" 
extension of the basic ARCH process, which does indeed utilize an ARMA 
process to model y^2(t).  See the discussion around and following 
Shumway and Stoffer's Eq. 5.45 or Zivot and Wang's Eq. 7.6

Regards,

   - Brian


> On 2/2/08, *Brian G. Peterson* <brian at braverock.com 
> <mailto:brian at braverock.com>> wrote:
> 
>     tom soyer wrote:
>      > Hi,
>      >
>      > I have a question with regard to different GARCH notations I
>     found in the
>      > literature, and I am wondering if anyone knows how to reconcile these
>      > differences. Below are three different notations that supposedly
>     all define
>      > the GARCH(1,1) process:
>      >
>      > In Zivot's book, MFTSWS, the GARCH(1,1) process is defined as:
>      > [1]: Y(t) = c + e(t), and
>      > [2]: sigma^2(t) = a0 + a1*e^2(t-1) + b1*sigma^2(t-1)
> 
>     I just looked in my current copy of Zivot and Wang MFTSwS+ (2006), p.
>     230, Eqs 7.4 and following, and your notation here doesn't match what's
>     in the reference (your Eq [2] appears equivalent to Eq. 7.5).  perhaps
>     next time you can be more specific in your reference (pages and Eq.
>     numbers?)
> 
> 
>      > In Engle's paper, the GARCH(1,1) process is defined (in financial
>     notation),
>      > like this:
>      >  [3]: r(t) = m(t) + sqrt(h(t))*e(t), and
>      > [4]: h(t+1) = a0 + a1*h(t)*e^2(t) + b1*h(t)
> 
>     I don't know which Engle paper you're referring to.  With the possible
>     exception of m(t) in your Eq[3] and the use of t+1 as the target in
>     Eq[4] (thus specifying the prediction), Eq [4] is equivalent to Eq [2]
>     and Eq [6]
> 
>      > In Stoffer's book, the GARCH(1,1) is define as:
>      > [5]: Y(t) = sigma(t)*e(t), and
>      > [6]: sigma^2(t) = a0 + a1*Y^2(t-1) + b1*sigma^2(t-1)
> 
>     Shumway and Stoffer "Time Series Analysis and Its Applications, 2nd
>     Ed."(2006), p. 286 Eqs. 5.30 and 5.44 match your Eq [5] and [6] and
>     match Zivot&Wang's representation.
> 
>     Note that Shumway and Stoffer also has several fairly extensive examples
>     of working with GARCH models in R.
> 
>      > Does anyone know if all three above are just different ways of
>     saying the
>      > same thing, or are they drastically different with respect to the
>      > specification of the GARCH model to be fitted?
> 
>     Notation is always a real pain to sort out as you are reading various
>     papers and books.  It is not uncommon to find errors in the references,
>     which is usually cleared up only via looking further back in time to
>     more primary sources.
> 
>     So, without precise references, I can only give you a qualified "these
>     models all appear equivalent".
> 
>     Regards,
> 
>       - Brian